Whenever I list my goals for any mathematics course I teach these days, the first goal on the list is to make my students into problem solvers. (Another goal is always to persuade them that learning mathematics can be enjoyable.) My emphasis on problem solving is not unusual; in fact, it was the first of the five “process standards” listed in NCTM’s Principles and Standards for School Mathematics and is the first of the eight “standards for mathematical practice” in the Common Core State Standards. Realistically, there are many parts of my mathematics courses that will be of little use to the students who do not go into STEM-related professions, but the ability to solve problems will serve all of them well throughout their lives.
You would think that “high-performing” students would be, almost by definition, great problem solvers, but I have tried to stress in previous blogs that this is highly dependent on how we measure high performance. A diligent student can be very good at showing us what we hope to see, but often because we have already demonstrated exactly what we hope to see before asking the student to show us. That diligent student might well earn an A in the course, and that is fair enough, since we should value diligence. However, we should also keep prodding our students to solve problems for which they have not been specifically prepped.
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Obviously, we should not hit our students with such questions for the first time in a graded environment like a test. They should, ideally, be solving problems in class each day so we can watch them in problem-solving action. In fact, if you can get them to solve a problem that stretches them a little bit beyond where they were the day before, then it becomes a great introduction to your new material. I recommend starting every day with a problem to be solved, taking full advantage of the time when students are most likely to be engaged. You can walk around and see how they are doing, and that can inform what you do next. (Let them go? Give a little hint? Rephrase the problem?) You can get them to show the solutions, reinforcing the idea that the teacher is not the only one who can do mathematics. If they are on a roll, you can add a new wrinkle to the problem (what if…) and see how they handle that. When all is said and done, you will probably have covered everything you would have done by showing them five stepped-out solutions, except this way the results will have come from them, and they will have been solving problems all along.
This kind of classroom dynamic is what NCTM has been advocating since the original Standards appeared more than 20 years ago, and the Common Core process standards are repeating that advocacy today. If we truly want our students to learn problem solving in our classrooms, there is really no other way to go. It is only after students have become accustomed to thinking their way through many problems without having seen identical problems beforehand that they will be ready to do the same thing on tests and quizzes (and, might I add, those high-stakes assessments). It will probably be necessary to adjust your expectations of student performance and to scale the raw grades (see my previous blogs), but eventually you will see a difference in their receptivity to new problems and their ability to solve them.
